Problem: All of the 4th grade teachers and students from Springer went on a field trip to an art museum. Tickets were $$5.00$ each for teachers and $$2.50$ each for students, and the group paid $$35.00$ in total. A few weeks later, the same group visited a science museum where the tickets cost $$20.00$ each for teachers and $$7.50$ each for students, and the group paid $$125.00$ in total. Find the number of teachers and students on the field trips.
Explanation: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${5x+2.5y = 35}$ ${20x+7.5y = 125}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-4$ ${-20x-10y = -140}$ ${20x+7.5y = 125}$ Add the top and bottom equations together. $ -2.5y = -15 $ $ y = \dfrac{-15}{-2.5}$ ${y = 6}$ Now that you know ${y = 6}$ , plug it back into $ {5x+2.5y = 35}$ to find $x$ ${5x + 2.5}{(6)}{= 35}$ $5x+15 = 35$ $5x = 20$ $x = \dfrac{20}{5}$ ${x = 4}$ You can also plug ${y = 6}$ into $ {20x+7.5y = 125}$ and get the same answer for $x$ ${20x + 7.5}{(6)}{= 125}$ ${x = 4}$ There were $4$ teachers and $6$ students on the field trips.